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Wet parts. The first thing you learn in ceramics is "score and slip." To attach 2 wet pieces of clay, you score both sides with a needle tool or fork, apply water or slip, and mush them together. However, some potters (even famous ones!) have abandoned the traditional score and slip method, saying they found direct contact to be more reliable. When using the direct contact method, it helps to have a binder. Vinegar works great for this, almost like a glue. Try it both ways and compare!
Dry parts. In general, you cannot attach two pieces of clay which have dried past the leather hard stage. However, there are some exceptions.
You can sometimes use a commercial mender such as Magic mender or APTII mender to attach pieces of greenware or even bisque. http://www.bigceramicstore.com/supplies/OtherThrowingHandbuilding
We have detailed instructions for using APTII here.
You can make your own potion. Lana Wilson, author of Ceramics: Shape and Surface, Handouts for Potters, uses this recipe called Magic Potion which she says is a super duper greenware fixer!!! The recipe is: I gallon water, 3 tablespoons liquid sodium silicate, 1 and 1/2 teaspoons of soda ash. *Liquid sodium silicate is also called egg keep and is sometimes available at drugstores.
Paper Clay is a mixture of clay and paper fibers. Using it you can attach wet and dry pieces, and they will not crack. There are several excellent books on Paper Clay. You can make it yourself by wedging blended paper pulp into clay, or buy it in 25# pugged bags. Paper Clay is excellent for handbuilding because of this ability to combine wet and dry pieces.
A final technique for encouraging good attachment of wet and dry clay is to wrap wet newspaper around the completed object, then wrap in plastic. This causes the moisture levels throughout the piece to even out, resulting in less cracking. Great for those darned feet that keep falling off that too dry pot. I recently submitted this to an extreme test, attaching wet clay to bone dry clay that I had "forgotten about" over the holidays. Worked great!